# Prove that the series SUM (-1)^n n/p_n converges where p_n are primes

1. Mar 5, 2007

### Dragonfall

1. The problem statement, all variables and given/known data

Prove that $$\sum(-1)^n\frac{n}{p_n}$$ converges, where $$p_n$$ is the nth prime.

2. Relevant equations

The sequence $$\frac{n}{p_n}$$ is definately not monotone if there exists infinitely many twin primes, since $$2n-p_n<0$$ for sufficiently large n, so alternating series test is out. Are there any other ways of showing this converges?

2. Mar 5, 2007

### StatusX

Can you use the prime number theorem? This says that:

$$\lim_{n \rightarrow \infty} \frac{p_n}{n \ln n} = 1$$

3. Mar 10, 2007

### Dragonfall

I can't use it for the series. I can only establish that n/p_n -> 0, which is insufficient for the series. I can't even prove that n/p_n is NOT monotone for large n, unless I assume the twin prime conjecture, for example.